Unbounded operators are mathematical objects used primarily in functional analysis and quantum mechanics. Unlike bounded operators, which have a finite limit on their output, unbounded operators can produce outputs that grow indefinitely. This means they can act on a larger set of functions, making them essential for describing physical systems where certain quantities, like momentum or position, can take on infinite values. In quantum mechanics, unbounded operators often represent observable quantities. For example, the position operator and the momentum operator are unbounded because they can yield results that extend to infinity. Understanding these operators is crucial for solving problems related to quantum states and wave functions.